The board / number theory / #853
Erdős Problem #853
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(d_n = p_{n+1} - p_n\), where \(p_n\) is the \(n\)th prime. Let \(r(x)\) be the smallest even integer \(t\) such that \(d_n = t\) has no solutions for \(n \le x\). Is it true that \(r(x) \to \infty\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/853.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/853
Attempts
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Certified every even consecutive-prime gap through 1572, proved \(r(92,489,057,256,472,962,493)\ge1574\), exactly computed \(r(50,847,533)=254\) with all prefix breakpoints, and reduced the two open l
PARTIAL